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Physical Structure of Matter 5.1.12-00 Electron spin resonance

Physics of the Electron

What you can learn about ...

Zeeman effect Energy quantum Quantum number Resonance g-factor Landé factor

Principle:

The g-factor of a DPPH (Diphenylpikrylhydrazyl) and the half-width of the absorption line are determined, using the ESR apparatus.

What you need:

ESR resonator with field coils ESR power supply Power supply, universal Oscilloscope, 30 MHz, 2 channels Digital multimeter Screened cable, BNC, l = 750 mm Adapter, BNC-socket/4 mm plug pair Connecting cord, l = 500 mm, blue Connecting cord, l = 500 mm, red Connecting cord, l = 500 mm, yellow Options: Teslameter, digital Hall probe, tangent., prot. cap 13610.93 13610.02 1 1 Electron spin resonance (ESR), model experiment. 09050.00 09050.93 13500.93 11459.95 07134.00 07542.11 07542.27 07361.04 07361.01 07361.02 1 1 1 1 1 4 1 3 2 2

Complete Equipment Set, Manual on CD-ROM included Electron spin resonance P2511200

Tasks:

With ESR on a DPPH specimen determination of 1. the g-factor of the free electron, and 2. the half-width of the absorption line.

212 Laboratory Experiments Physics

PHYWE Systeme GmbH & Co. KG · D - 37070 Göttingen

Electron spin resonance

LEP 5.1.12 -00

Related topics Zeeman effect, energy quantum, quantum number, resonance, g-factor, Landé factor. Principle The g-factor of a DPPH (Diphenylpikrylhydrazyl) and the halfwidth of the absorption line are determined, using the ESR apparatus. Equipment ESR resonator with field coils ESR power supply Power supply, universal Oscilloscope, 30 MHz, 2 channels Digital multimeter Screened cable, BNC, l = 750 mm Adapter, BNC-socket/4 mm plug pair Connecting cord, l = 500 mm, blue Connecting cord, l = 500 mm, red Connecting cord, l = 500 mm, yellow Options: Teslameter, digital Hall probe, tangent., prot. cap

Tasks With ESR on a DPPH specimen determination of 1. the g-factor of the free electron, and 2. the half-width of the absorption line.

09050.00 09050.93 13500.93 11459.95 07134.00 07542.11 07542.27 07361.04 07361.01 07361.02

1 1 1 1 1 4 1 3 2 2

Set-up and procedure A symmetrically fed bridge circuit (Fig. 2a) contains a variable resistor R in one branch and a high-quality tuned circuit (resonator) in the other. The specimen is located in the coil of the tuned circuit. Normally, the bridge is balanced so that the complex impedace of both branches is the same and consequently there is no voltage between points a and b. If the external magnetic field is now so adjusted that the resonance absorption occurs in the specimen, the bridge becomes unbalanced and the voltage set up between a and b rectified and amplified. If the magnetic field is modulated with 50 Hz a.c. (voltage 2 V), the resonance point is passed through 100 times a second (Fig. 3), and the absorption signal can be displayed on an oscilloscope, provided the x-deflection is driven with the same a.c. voltage in the correct phase.

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Fig. 1: Experimental set-up for determining characteristic curves.

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Fig. 2a: Measuring bridge of the ESR apparatus.

Fig. 3: The magnetic field B is compounded from a d.c. field B= and an alternating field B , so that B = B= + B . Through l=, B= is to be adjusted so that B= = Br.

First, the bridge has to be balanced. In doing this (without external magnetic field), "R" on the resonator is brought to its central position and "C" to the lefthand stop. On the ESR power supply, key 8 "bridge balancing" (see operating instructions) is pressed, the oscilloscope input is switched to d.c. and 1 V/cm and the line is brought to the zero with 12 "Zero" (beforehand bring to zero with "Position" in GND setting). For the wiring see Fig. 2b and operating instructions. Now the absoption signal can be searched for: the coil current is set to about 1.3 A, button 10 " " pressed and the zero line again brought to the centre with 12. The signal can now be sought with 25 "C", while continuously correcting the zero point with 12. As soon as a signal appears, both lines are made to coincide with 13 "Phase". As long as the resonance frequency of the tuned circuit does not correspond to the oscillator frequency, the signal appears asymmetrical or as a derivative. The resonator should thus be tuned with 25 "C" until a symmetrical absorption signal of maximum amplitude appears on the oscilloscope. The sensitivity of the oscilloscope is increased so that the signal reaches a height of 8-10 cm. By varying the d.c. current in the coil, the signal can be brought to the centre of the screen, so that the minimum is on the y-axis, (correct again with 13 "Phase" if necessary). The resonance magnetic field Br can be calculated from the d.c. current lres now flowing. Fig. 2b: Wiring diagram. If the signal is now adjusted so that the x-axis goes exactly half way through the height of the signal, then, once the alter-

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PHYWE series of publications · Laboratory Experiments · Physics · © PHYWE SYSTEME GMBH & Co. KG · D-37070 Göttingen

Electron spin resonance

LEP 5.1.12 -00

nating voltage modulation has been switched off, the two current values at which the moving spot crosses the x-axis can be determined by slowly varying the d.c. current in the coil. The half-width of the signal is calculated from the difference between these currents.

In this case, HZ (2) can be written HZ = mB gj B · J with the Landé factor gj 1 J 1J 12 2J 1J S 1S 12 12 L 1L 12 . (4) (3)

Theory and evaluation If an electron of mass m and charge e is located in an electromagnetic field with vector potential A and scalar potential f, then its steady-state energy level and characteristic states are given by the solutions of the Dirac equation. If the equation is solved for the c, component of the sate spinor, it reads:

Disregarding the influence of the nuclear spin, the energy levels of HZ, for magnetic fields in the Z-direction, are EZ = mB gj B mj with mj = j, j 1, ... j. The selection rule for magnetic transitions of the electron spin resonance experiment is mj = ±1 so that the absorption condition reads mB gj B = E = hf.

S 1 p2

where mB = S p

S S mB 2 S · B 2 p1

eh 2mc

ec1

(1)

(Bohr Magneton) Am2

9.27 · 1027 S p S e>c A

where p is the momentum, c the velocity of light, S the electron spin operator, and B = rot A is the magnetic field. h is Planck's constant h = 6.626 · 1034 Js, and e 1 11 2 1E>mc2 2 2 1E mc2 2

In many cases, especially with molecular radicals, the angular momentum L of the unpaired electron is extinguished by the electric fields of the neighbouring atoms and molecules. In the case of DPPH, L = 0 and therefore gj = 2. In (3), m = mB · gj · J (5)

is approximately the excess of energy over the residual mass energy. For a free electron in a uniform magnetic field, the second term (Zeeman term) interchanges with the first in the Hamiltonian operator of (1), and the energy level e = eo + mB 2 Sz Bz, is obtained

is the magnetic moment of an electron with the angular momentum J in units of the Bohr magneton mB . If account is also taken of the exchange of virtual photons between the electron and a radiation field in the static limit through inclusion of the vertex corrections in increasing order, a modified abnormal magnetic moment of the electron is obtained as a series in the fine-structure constant a: a e2 h·c mB a 1 a 2p 0.328 a2 p2 S ... b gj J (6)

if the z-axis lies in the direction of the magnetic field. eo is the energy of the electron without a magnetic field. If the diamagnetic contribution from the first term of (1) is disregarded in the general case, then for the electron in a uniform magnetic field, the interaction Hamiltonian operator with the magnetic filed, (Zeeman effect) is HZ = mB B · ( L + 2 S ), (2)

S m

where L and S are the operators of the orbital and spin angular momentum respectively. In addition, the spinorbit interaction Hso = l · S · L should be taken into account, so that only the total angular momentum J = L + S is a conservation value.

The correction factor in the parentheses of equation (6) is generally taken into account in gj, so that gj then 2. Fig. 4 shows the energy level in the magnetic field B, for DPPH. By supplying the corresponding energy, transitions can be induced between the levels: hf = gj mB H. The probability of transtion depends on the occupation number and on the transition matrix elements. The latter are the same for absorption and emission processes. Because of the interactions of the spins with the lattice or with one another, the levels are not sharply defined, and this leads

PHYWE series of publications · Laboratory Experiments · Physics · © PHYWE SYSTEME GMBH & Co. KG · D-37070 Göttingen

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to a line width of the absorption spectrum and prevents an equipartition of the levels (saturation) because of the corresponding relaxation processes. The occupation numbers are given in accordance with the Boltzmann relation by: N2 N1

¢E gjmBB

For the g-factor, one obtains: g = 10.43 · or g = 2.507 · 1 ; Ir in amperes, Ir 1 ; Br in teslas Br

e

kT

e

kT

,

with the stated values. A resonance current of 1.245 A was measured, corresponding to a g-factor of 2.02, and a half-width value of 2.7 · 104 T. (Literature values for DPPH: g = 2.0037, half-width 2.8 · 104 T). Note The analyzed specimen DPPH (Diphenylpicrylhydrazyl) is a organic, paramagnetic material with one stable radical. The magnetic moment of the molecul is determined only by the spin moment of the valence in the "N"-bridge.

where k is the Boltzmann constant. It is customary to take account of other interactions in the constant g which vary linearly B, e.g. the contact of Fermi interaction (WW) with the nucleus, i.e. g = gj (1 + WW), where g is specific for the measured substance and f is predetermined at 146 MHz by the apparatus, so that B must be varied to satisfy the resonance condition. For the magnetic field of a Helmholtz arrangement with w = 241 turns, current I and radius R = 0.048 m, B = 0.6445 · mo · w·I R

on the axis of symmetry between the coils, where mo = 1.256 · 106 · T·m A

With the numerical values stated, one obtains B = 4.0642 · 103 · i T . A

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Electron spin resonance (ESR), model experiment Model experiment for electron spin resonance for clear demonstration of interaction between the magnetic moment of the electron spin with a superimposed direct or alternating magnetic field. A ball with a central rod magnet, which rotates with low friction on an air cushion, acts as model electron. Two pairs of coils allow to generate a direct magnetic field Bo and an alternating magnetic field B1. The axes of both fields intersect perpendicularly at the centre of the ball. The table is slightly inclined to start the electron gyroscope with an air draught (Magnus effect). If the direct magnetic field Bo acts on the ball, a precession of the magnet axis is observed. Precession frequency increases with the intensity of field Bo. With a second pair of coils and a pole changeover switch (commutator), a supplementary alternating field B1 is generated. If the change of poles occurs at the right phase, the angle between the gyroscope axis and the direction of the direct field is continuously increased, until the magnetic axis of the ball is opposed to the field direction (spin flip). Ask for a complete equipment list Exp.no. P1298000

Material Gyroscope w. magnetic axis, ESR model Coil 1200 turns Iron core, short, laminated Commutator switch On/off switch Blower Pressure tube with end piece 1.5 m Variable transformer with rectifier 25 V~/20 V, 12 A Connecting cable

11208.00 06515.01 06500.00 06034.03 06034.01 13770.93 11205.01 13531.93

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Fig. 4: Electron spin resonance (ESR), model experiment.

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